Absorption

Absorption is the physical process that reduces the spectral radiance of a beam of light as it passes through a medium. It can be described be Beer’s law:

\[\vec{I}(r) = \vec{I}(0) \exp(-\mathbf{K} r),\]

where \(\vec{I}(r)\) is the spectral radiance as a Stokes vector of the light at some distance \(r\), and \(\mathbf{K}\) is the propagation matrix of the medium.

The unit of spectral radiance is W sr \(^{-1}\) m \(^{-2}\) Hz \(^{-1}\). The unit of the propagation matrix is m \(^{-1}\).

The Stokes Vector

In ARTS, the spectral radiance is represented by the Stokes vector, which is a 4-long column vector.

\[\begin{split}\vec{I} = \left[ \begin{array}{c} I_I \\ I_Q \\ I_U \\ I_V \end{array} \right],\end{split}\]

where \(I_I\) is the total spectral radiance, \(I_Q\) is the difference in spectral radiance between horizontal and vertical linear polarizations, \(I_U\) is the difference in spectral radiance between plus 45 and minus 45 linear polarizations, and \(I_V\) is the difference in spectral radiance between right and left circular polarizations. The unit of all of these is W sr \(^{-1}\) m \(^{-2}\) Hz \(^{-1}\).

This is the same definition as, e.g., Mishchenko et al. [4].

Tip

Because of the nature of circular polarization, it is not uncommon to see different \(I_V\) definitions in different context. Be sure to check the definition of \(I_V\) in the context you are working in. It has caused confusion in the past, for both users and developers of ARTS.

The Stokes Vector Unit Conversion

All ARTS simulations expect the incoming spectral radiance to be in W sr \(^{-1}\) m \(^{-2}\) Hz \(^{-1}\).

However, several ways to convert the Stokes vector to other units are available. These are listed below in no particular order.

Rayleigh Jeans Brightness Temperature

As the spectral temperature of the Rayleigh Jeans interpretation of black body spectral radiance. The conversion is given by

\[\begin{split}\vec{I}_{RJBT} = \frac{c^2}{2k\nu^2} \left[ \begin{array}{c} I_I \\ I_Q \\ I_U \\ I_V \end{array} \right],\end{split}\]

where \(k\) is the Boltzmann constant, \(c\) is the speed of light, and \(\nu\) is the frequency.

The unit of all of these is K.

Planck Brightness Temperature

As the spectral temperature of the black body spectral radiance. The conversion is given by

\[\begin{split}\vec{I}_{Tb} = \frac{h \nu}{k} \left[ \begin{array}{lcr} \log\left(1 + \frac{2h\nu^3}{c^2I_I}\right)^{-1} &&\\ \log\left(1 + \frac{4h\nu^3}{c^2(I_I + I_Q)}\right)^{-1} &-& \log\left(1 + \frac{4h\nu^3}{c^2(I_I - I_Q)}\right)^{-1} \\ \log\left(1 + \frac{4h\nu^3}{c^2(I_I + I_U)}\right)^{-1} &-& \log\left(1 + \frac{4h\nu^3}{c^2(I_I - I_U)}\right)^{-1} \\ \log\left(1 + \frac{4h\nu^3}{c^2(I_I + I_V)}\right)^{-1} &-& \log\left(1 + \frac{4h\nu^3}{c^2(I_I - I_V)}\right)^{-1} \end{array} \right],\end{split}\]

where \(k\) is the Boltzmann constant, \(h\) is the Planck constant, \(c\) is the speed of light, and \(\nu\) is the frequency.

The unit of all of these is K.

The Spectral Radiance per Wavelength

As the spectral temperature of the Rayleigh Jeans interpretation of black body spectral radiance. The conversion is given by

\[\begin{split}\vec{I}_{\lambda} = \frac{\nu^2}{c} \left[ \begin{array}{c} I_I \\ I_Q \\ I_U \\ I_V \end{array} \right],\end{split}\]

where \(c\) is the speed of light, and \(\nu\) is the frequency.

The unit of all of these is W sr \(^{-1}\) m \(^{-2}\) m \(^{-1}\).

The Spectral Radiance per Kayser

As the spectral temperature of the Rayleigh Jeans interpretation of black body spectral radiance. The conversion is given by

\[\begin{split}\vec{I}_{cm} = \frac{1}{c} \left[ \begin{array}{c} I_I \\ I_Q \\ I_U \\ I_V \end{array} \right],\end{split}\]

where \(c\) is the speed of light.

The unit of all of these is W sr \(^{-1}\) m \(^{-2}\) m.

Propagation Matrix

The propagation matrix conceptually describes how the Stokes vector propagates through a system. The propagation matrix is a square matrix with strict symmetries, and it has the form

\[\begin{split}\mathbf{K} = \left[ \begin{array}{rrrr} K_A & K_B & K_C & K_D \\ K_B & K_A & K_U & K_V \\ K_C &-K_U & K_A & K_W \\ K_D &-K_V &-K_W & K_A \end{array} \right],\end{split}\]

where \(K_A\) describes the total power reduction, \(K_B\) describes the difference in power reduction between horizontal and vertical linear polarizations, \(K_C\) describes the difference in power reduction between plus 45 and minus 45 linear polarizations, \(K_D\) describes the difference in power reduction between right and left circular polarizations, \(K_U\) describes the phase delay between right and left circular polarizations, \(K_V\) describes the phase delay between plus 45 and minus 45 linear polarizations, and \(K_W\) describes the phase delay between horizontal and vertical linear polarizations. The unit of all of these is m \(^{-1}\).

In practice, the propagation matrix is a sum of multiple physical processes:

\[\mathbf{K} = \sum_i \mathbf{K}_i\]

where \(i\) is the pseudo-index of the physical process. A physical process here refers to line-by-line absorption, collision-induced absorption, Faraday rotation, scattering, etc. Think of this sum as if each contribtion the the propagation matrix is completely independent of the others.

Sources of Absorption

The different ways to compute different \(\mathbf{K}_i\) are described in these sections: